# Cleaning Robot

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Dr. Asimov, a robotics researcher, loves to research, but hates houseworks and his house were really dirty. So, he has developed a cleaning robot.

As shown in the following figure, his house has 9 rooms, where each room is identified by a letter of the alphabet:

The robot operates as follows:

• if the battery runs down, the robot stops;
• if not so, the robot chooses a direction from four cardinal points with equal probability, and moves to the room in that direction. Then, the robot cleans the room and consumes 1 point of the battery;
• However, if there is no room in that direction, the robot does not move and remains in the same room. In addition, there is a junk room in the house where the robot can not enter, and the robot also remains when it tries to enter the junk room. The robot also consumes 1 point of the battery when it remains in the same place.

A battery charger for the robot is in a room. It would be convenient for Dr. Asimov if the robot stops at the battery room when its battery runs down. Your task is to write a program which computes the probability of the robot stopping at the battery room.

You are given battery capacity `cap` and `start, bat, junk` respectively represent the room where the robot is initially, the battery room, and the junk room.

Input: One integer (int) and three strings (str).

Output: Float (float).

Examples:

```assert cleaning_robot(1, "E", "A", "C") == 0.0
assert cleaning_robot(1, "E", "B", "C") == 0.25
assert cleaning_robot(2, "E", "A", "B") == 0.0625
```

This mission was taken from AIZU ONLINE JUDGE

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